# Questions tagged [sheaf-cohomology]

The sheaf-cohomology tag has no usage guidance.

260
questions

**5**

votes

**1**answer

165 views

### Elementary proof of the exactness of Čech complex associated to a hypercovering (“Illusie's Conjecture”)

Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...

**3**

votes

**0**answers

61 views

### Sheaf cohomology of Grassmannian G(2,4) with values in twisted tautological bundles over an arbitrary field

Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to ...

**12**

votes

**0**answers

414 views

### Is there a concrete application of topos theory?

The notion of topos was originally formulated in SGA 4 in the context of attacking the Weil conjectures. This formalism turned out to be unnecessary for the purposes of proving those conjectures. But ...

**2**

votes

**1**answer

191 views

### Help about “Varieties with small Dual Varieties” by L.Ein

I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...

**5**

votes

**1**answer

129 views

### Cohomology of doubly pinched torus via spectral sequences

Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...

**3**

votes

**0**answers

184 views

### Spectral sequence from resolution of condensed abelian groups

I am watching Scholze's and Clausen's masterclass on Condensed Mathematics and I don't understand or can find any references on something they said.
You have a resolution
$$ \dots \to \mathbb{Z}[\...

**1**

vote

**1**answer

229 views

### Relationship between $H^1(X, \mathbb{T})$ and complex line bundles

Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, ...

**3**

votes

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166 views

### Sheaf cohomology on the formal completion of $\mathbb{P}^1_k$ at its north pole

Let $k$ be a field and let $\mathbb{P}^1_k$ be the projective line over $k$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^1_k$. The curve $\mathbb{P}^1_k$ is recovered by two affine subschemes ...

**0**

votes

**0**answers

137 views

### Dimension of global holomorphic sections of a line bundle

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...

**3**

votes

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99 views

### Elementary reference for Borel-Moore/locally finite homology

There is a homology theory called "Borel-Moore" or "locally finite" homology, which can either be constructed by using locally-finite chains or by more advanced sheaf-theoretic ...

**1**

vote

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57 views

### $L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality

This is a question arising when I am reading
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
...

**1**

vote

**0**answers

107 views

### Sections of the structure sheaf of a partial flag variety on big cell

Let $G$ be a connected split reductive group over a (non-archimedean) field $K$ of char 0 with split maximal torus $T$ and a standard parabolic $P$. Denote by $W$ the Weyl group of $G$ and by $W_P$ ...

**4**

votes

**0**answers

124 views

### Continuity property for Čech cohomology

Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...

**0**

votes

**0**answers

43 views

### Image of first group of Cech cohomology in the second group of De Rham cohomology

Let $M$ be a smooth manifold and $\mathcal{U}$ an open good cover of $M$. If $\underline{\mathbb{Z}}$ denotes sheaf of locally constant functions, $C^{\infty}(U) := C^{\infty}(U, \mathbb{R})$ and $\...

**2**

votes

**0**answers

104 views

### Are flasque sheaves exactly the retracts of “canonically” flasque sheaves?

Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ ...

**1**

vote

**0**answers

133 views

### Explicit map between $\check{H}^1(M,\underline{\mathbb{R}})$ and $H^1(M,\mathbb{R})$

Is there a way to construct an explicit isomorphism between Cech cohomology and singular cohomology on a smooth manifold for degree 1? If yes can this be extended to higher degee?

**0**

votes

**1**answer

93 views

### Connecting homomorphism in Cech cohomology

Let $M$ be a smooth manifold and $\mathcal{U}$ be a good open cover of $M$. If I have an exact sequence of sheaves
$$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \...

**4**

votes

**0**answers

175 views

### Does cohomology and base change hold if supported at a point?

I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$.
Q1: Does "cohomology and ...

**3**

votes

**1**answer

265 views

### Elementary way to compute Hodge numbers of Grassmanian

I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf ...

**8**

votes

**0**answers

390 views

### intuition about perverse sheaves

firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...

**1**

vote

**0**answers

90 views

### Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...

**4**

votes

**0**answers

70 views

### Serre vanishing on one-point blow-ups

This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry.
Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...

**1**

vote

**1**answer

107 views

### Relation between characteristic cycle and singular support of constructible sheaf

Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $...

**1**

vote

**0**answers

63 views

### Compute Cech cohomology with two open sets

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...

**4**

votes

**1**answer

202 views

### Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$

I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\...

**1**

vote

**0**answers

140 views

### Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)

I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2
& Example 12.9.2):
Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point
$(0:...:1)...

**4**

votes

**2**answers

292 views

### Sheaf cohomology commutes with colimits of sheaves

Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...

**5**

votes

**0**answers

245 views

### Calculation in prismatic cohomology

In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper ...

**12**

votes

**1**answer

283 views

### Sheaves in combinatorics and discrete geometry

I am looking for examples for the application of sheaves, sheaf-like constructions or the (co)homology of sheaves to problems in combinatorics and discrete geometry.
For example given a poset $(P,\...

**1**

vote

**0**answers

111 views

### Surjectivity of multiplicative map (in more specific case)

(I have asked the question Surjectivity of multiplicative map. I ask here the more specific case.)
Let $S$ be a smooth complex algebraic surface, and $D$ be a divisor on $S$ such that $D^2>0$ and $...

**0**

votes

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124 views

### Surjectivity of multiplicative map

Let $S$ be a smooth complex algebraic surface, and $\mathcal{F}$ be a coherent sheaf on $S$.
I want to consider $W = S \times S$ and the coherent sheaf $\mathcal{G} = \mathcal{F} \boxtimes \mathcal{F}...

**1**

vote

**1**answer

77 views

### Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes.
Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$.
We can define the subsheaf $\...

**4**

votes

**0**answers

101 views

### Understanding a step in proof of sheaf version Verdier duality

Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss.
So all proofs I can find factors through a particular statement, which goes ...

**3**

votes

**1**answer

98 views

### Subspace inclusion with non-vanishing higher direct images

I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...

**1**

vote

**0**answers

42 views

### local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$.
Suppose that $K$ is locally ...

**2**

votes

**0**answers

109 views

### A infinity structure on Yoneda Ext group

I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...

**1**

vote

**1**answer

176 views

### Injectivity of the cohomology map associated to the pullback of line bundles

Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...

**1**

vote

**0**answers

265 views

### Čech-Alexander complex in computing (crystalline/prismatic) cohomology

I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology).
They seemed to be introduced as a method ...

**3**

votes

**0**answers

141 views

### Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology

There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...

**1**

vote

**1**answer

346 views

### Pullback map on global sections surjective

Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ ...

**1**

vote

**0**answers

114 views

### Group cohomology of sheaves under closed immersion

Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...

**4**

votes

**1**answer

351 views

### The Yoneda pairing, hypercohomology, and cup product

Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...

**2**

votes

**1**answer

311 views

### Very weak Riemann-Roch on curves (by J. Kollar)

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on curves)...

**3**

votes

**1**answer

285 views

### Does local cohomology commute with pullback?

Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open.
For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...

**8**

votes

**0**answers

217 views

### Global functions on a product of schemes over artinian ring

For a morphism of schemes $f:X\to S$ with $S=\text{Spec}(R)$ affine, let's write $A(X)=H^0(X,\mathcal{O}_X)$. I'm interested in the morphism of $R$-algebras
$$
c:A(X)\otimes_R A(Y)\to A(X\times_SY)
$$
...

**3**

votes

**0**answers

144 views

### Sheaf cohomology of the complement of a schubert variety

Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...

**10**

votes

**3**answers

933 views

### How to compute the cohomology of a local system?

Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well.
Suppose that we are ...

**2**

votes

**1**answer

207 views

### Pushforward in Compactly Supported Cohomology

Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...

**8**

votes

**1**answer

287 views

### Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?

Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...

**1**

vote

**1**answer

233 views

### Multiplicative structure for sheaf cohomology of flag varieties

Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...